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Suppose that $f:\mathbb R^2\to\mathbb R_+$ is a non-negative continuous function and that $$\int_{y\in\mathbb R}\int_{x\in\mathbb R}f(x,y)\,\mathrm dx\,\mathrm dy<\infty.$$ Is it true that $$\int_{x\in\mathbb R}f(x,y)\,\mathrm d x<\infty$$ for all $y\in\mathbb R$, as opposed to merely almost all $y\in\mathbb R$?

My hunch is that the answer is affirmative given the continuity of $f$, but I can come up with neither a proof nor a counterexample. Any hint would be appreciated.

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This is false. Consider the function $$ f(x,y) = \exp\left(-\left|\sqrt{\smash{|y|}\vphantom{L}} \cdot x\right|\right) \cdot (1 - y^2) \cdot 1_{(-1,1)}(y). $$

This is continuous and integrable, but $f(x , 0) \equiv 1$ is not integrable.

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  • $\begingroup$ Great counterexample, thank you! $\endgroup$ – triple_sec Jan 7 '16 at 9:31

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